Further Mathematics Unit 3 Page 1 Week 11SEND Work SEND: Work for Submission 11 (2022) 1. Tom breeds birds. He started with 10 000 birds. The bird population increases on average 20% per year. a) Suggest a recurrence relation model we can use to predict the bird population after each year. b) Use the model to predict the bird population after 7 years. c) Give an estimate of when the bird population can be expected to exceed 50 000. 2. If $4 200 is invested earning 7% per annum in interest compounded annually, a) Write a recurrence relation to model this investment. b) Use the recurrence relation to find the value of the investment after 1, 2, 3 and 4 years. c) Determine how many years it takes for the value of the investment to first exceed $10 000. d) How much interest was earned just in the third year of the investment? 3. Jane opened a saving accounts with Bank P. Let P. be the balance of the savings account " years after it was opened. The value of P. can be determined using the recurrence relation model below. Po=A Pal = 1.056 x P The balance of the savings account one year after it was opened was $1584. a) Find the value of A. b) What is the annual interest rate of this saving accounts? c) Write down the balance of the savings account four years after it was opened. 4. Office furniture was purchased new for $18 000. It will be depreciated using a reducing-balance depreciation method with an annual depreciation rate of 4.5%. Let I'm be the value of the furniture after n years. a) Write a recurrence relation to model the value of the furniture from year to year. b) Generate a sequence of numbers that represent the value of the furniture from year to year for 5 years in total. Write the values of the terms of the sequence correct to the nearest cent. c) What is the value of the furniture after 3 years? d) What is the total depreciation of the furniture after 5 years?Further Mathematics Unit 3 Page 2 Week 1 1 SEND Work 5. Roland is a keen cyclist who hopes to become a professional cyclist. He purchased a quality road bike for training and competitions. The road bike was purchased for $6 640 and could be depreciated using the reducing balance depreciation method with the recurrence relation Vo = 6 640 Vn+1 = 0.852 x Vm where I'm is the value, in dollars, of the road bike after n years. a) Using the recurrence relation, show that the value of the road bike after two years is $4820, correct to the nearest dollar. b) State the annual reducing balance depreciation rate for Roland's road bike. c) Roland plans to sell the road bike when its value drops below $1300. When would he sell the road bike if he follows this plan? 6. The graph below shows the value, I'm, of an asset as it depreci es over a period of five months. value ($) n (months) 5 Which one of the following depreciation situations does this graph represent? Explain your choice. A. Flat rate depreciation with a decrease in depreciation rate after two months B. Flat rate depreciation with an increase in depreciation rate after two months C. Unit cost depreciation with a decrease in units per month after 2 months D. Reducing balance depreciation with an increase in the rate of depreciation after 2 months. E. Reducing balance depreciation with a decrease in the rate of depreciation after 2 monthsFurther Mathematics Unit 3 Page 3 Week 1 1 SEND Work Week 11 Online Quiz Every week there are five multiple-choice questions to attempt online as part of the work submission. Log in to DECV online, select Further Maths and click the link for Week 11 Quiz. Restrict your time to 10 minutes only. Below are the quiz questions. You can do them first and then go online to enter your responses. If you get less than 4/5, this indicates that you need to spend more time reviewing the work for the week. Circle the letter beside the correct answer. 1. The sequence -2, 6, -18, 54, -162, ... can be defined by the first-order recurrence relation: A Wall = -21. . =-2 But = -2u, .. 1 =3 C umHI = -31. . M = 3 D un= -3u. . W=-2 Eu, = 31. . W, =2 2. The first four terms of a sequence are plotted on the graph shown. Which of the following first-order recurrence relations could describe the sequence? 1 = 2 Un A 12 B U.. = 21. 1 = 1 10 Value of term D HAH = 3U. 1 =1 0 2345 E U. = 31. 1 = 2 Term numberFurther Mathematics Unit 3 Page 4 Week 1 1 SEND Work 3. The first 6 terms of the following recurrence relation are: 1 = 1 A 1,-3. -7,-11, -15, -19 B -1, -4, -16, -64, -256, -1024 C 1, -4, 16, -64, 256, -1024 D -1,4, -16, 64, -256, 1024 E 1, 4, 16, 64, 256, 1024 4. Harry is breeding trout in a pond. He fills his pond with 700 trout and expects that the number of trout will increase by 9% each month. After 1 year, the number of trout in Harry's pond is expected to be: A 1968 B 2146 C 1806 D 1657 E 2339 5 $7000 is invested earning 10% in interest compounded yearly. The recurrence relation used to model this investment is: A Vo = 7000 Vn+1 = 0.1Vm B Vo = 7000 Vn+1 = 1.1Vm C Vo = 7000 Vn+1 = 10Vr D Vo = 7000 Vn+1 = 7000VE E Vo = 7000 Vn+1 = 70.1Vm